Most of the talk is based on joint work with my former PhD students: Kyle Austin and Michael Holloway.
Its purpose is to investigate the duality between large scale and small scale. It is done by creating a connection
between C*-algebras and scale structures.

In the commutative case we consider C*-subalgebras of $C^b(X)$, the C*-algebra of bounded complex-valued functions on $X$. Namely, each C*-subalgebra $\mathscr{C}$ of $C^b(X)$ induces both a small scale structure on $X$ and a large scale structure on $X$. The small scale structure induced on $X$ corresponds (or is analogous) to the restriction of $C^b(h(X))$ to $X$, where $h(X)$ is the Higson compactification.

The large scale structure induced on $X$ is a generalization of the $C_0$-coarse structure of N.Wright. Conversely, each small scale structure on $X$ induces a C*- subalgebra of $C^b(X)$ and each large scale structure on $X$ induces a C*-subalgebra of $C^b(X)$. To accomplish the full correspondence between scale structures on $X$ and C*-subalgebras of $C^b(X)$ we need to enhance the scale structures to what we call hybrid structures. In the noncommutative case we consider C*-subalgebras of bounded operators $B(l_2(X))$.

We consider the problem of classification of signals. Through the use of delay embedding and persistent homology, we transform signals into the space of persistence diagrams. We introduce a distance on this space inspired by a distance used in the fields of point processes and multi-set filtering problems. Using this distance, we propose classification and clustering schemes on the persistence diagrams, which is then benchmarked on real and synthetic data.

Graphene is a material made of carbon atoms arranged in a 2-dimensional hexagonal lattice. It was first produced in sheets at room-temperature in 2004 and has since become a widely studied subject in materials science and physics due to it being both the strongest and most conductive material known to man. I will demonstrate the use of crystal
geometry methods to calculate the anisotropic edge energy of graphene as a function over all commensurate edge orientations and that the Wulff shape of this function correctly corresponds with graphene's equilibrium shape. I will also present a result on the energy of non-commensurate edges. These methods can then be used to describe the
edge/surface energies for a wide variety of crystals with non-Bravais lattice structure.

I will review the notions associated with traditional tilings of the plane, conformal tilings, subdivision rules, and aggregate tiles. This is in preparation for a second talk in which I'll prove the convergence of aggregate conformal tiles to their traditional tile shapes.

In light of the Ebola outbreak in 2014, we worked on an Ebola model during our South Africa Mathematical Sciences Association Masmau program in 2014 and 2015. Our model partitions the population into those who take precautions against contracting the disease and those who do not. We consider new infections arising in both hospital settings as well as in the community, and include transmission from dead bodies and the environment. Our goal is to illustrate role of education in limiting a potential future Ebola outbreaks in Sudan using data and modeling. We considered implications of a new strain with respect to different death rates and recovery rates.

We introduce a new, reduced order NS-$\alpha$ (rNS-$\alpha$) model for the purpose of efficient, stable and accurate simulations of incompressible flow problems at high Reynoldsnumbersoncoarsemeshes. Wemotivatethenewmodelasanadaptationofthe well-known NS-$\alpha$ model that is more efficiently computable, then analyze its well- posedness,treatmentofenergy,anddiscussnumericaldiscretizations. Severalnumerical tests are given which reveal remarkable coarse-mesh accuracy for turbulent flow simulations. Finally, we examine sensitivity of the models solutions to the filtering radius.

If you are interested in giving or arranging a talk for one of our seminars or colloquiums, please review our calendar.

If you have questions, or a date you would like to confirm, please contact colloquium AT math DOT utk DOT edu